example 1.7\nif the marginal cost of producing $x$ units of a commodity is given by\n$c(x)=0.3x^{2}+2x$\nand…

Explanation:

Step1: Integrate marginal cost

The marginal cost $C'(x) = 0.3x^{2}+2x$. Integrating to find the cost - function $C(x)=\int(0.3x^{2}+2x)dx$. Using the power - rule of integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $C(x)=0.3\times\frac{x^{3}}{3}+2\times\frac{x^{2}}{2}+K$, where $K$ is the constant of integration. Simplifying gives $C(x)=0.1x^{3}+x^{2}+K$.

Step2: Determine the constant $K$

The fixed cost is the cost when $x = 0$. Given that the fixed cost is $$2000$, when $x = 0$, $C(0)=2000$. Substituting $x = 0$ into $C(x)=0.1x^{3}+x^{2}+K$, we get $C(0)=K$. So $K = 2000$, and the cost function is $C(x)=0.1x^{3}+x^{2}+2000$.

Step3: Calculate cost of 20 units

Substitute $x = 20$ into the cost function $C(x)$. $C(20)=0.1\times(20)^{3}+(20)^{2}+2000$. First, calculate $0.1\times(20)^{3}=0.1\times8000 = 800$, $(20)^{2}=400$. Then $C(20)=800 + 400+2000=3200$.

Answer:

The cost function is $C(x)=0.1x^{3}+x^{2}+2000$, and the cost of producing 20 units is $$3200$.